A collage-based approach to inverse problems for non-linear elliptic PDEs

The collage coding solution strategy for ODE inverse problems, first developed in Kunze and Vrscay (1999), controls the approximation error by bounding it above by a more readily minimisable distance. In this paper, we explore a collage coding technique for solving inverse problems for a general class of second-order non-linear elliptic PDEs. The method is based on the nonlinear Lax-Milgram representation theorem and elements of variational calculus. We develop this method for a broad class of non-linear elliptic PDEs and present an example of the method in practice.